Plotting & Interpreting Multiple Regression Models• Why plot MR models?• Interpretive review & extensions • Plotting single-predictor models– centered quant, binary or k-category predictors• Plotting two-predictor models– centered quant & binary, k-category or centered quant• Plotting multiple-predictor modelsSo far we have emphasized the clear interpretation of regressionweights for each type of predictor.Many of the “more complicated” regression models, such as ANCOVA or those with non-linear or interaction terms are plotted, because, as you know, “a picture is worth a lot of words”.Each term in a multiple regression model has an explicit representation in a regression plot as well as an explicit interpretation – usually with multiple parallel phrasing versions. So, we’ll start simple with simple models and learn the correspondence between:• interpretation of each regression model term• the graphical representation of that termCoding & Transforming predictors for MR models• Categorical predictors will be converted to dummy codes• comparison/control group coded 0• @ other group a “target group” of one dummy code, coded 1• Quantitative predictors will be centered, usually to the mean• centered = score – mean (like for quadratic terms)• so, mean = 0Why?Mathematically – 0s (as control group & mean) simplify the math & minimize collinearity complicationsInterpretively – the “controlling for” included in multiple regression weight interpretations is really “controlling for all other variables in the model at the value 0”– “0” as the comparison group & mean will make b interpretations simpler and more meaningfulVery important things to remember…1) We plot and interpret the model of the data -- not the data• if the model fits the data poorly, then we’re carefully describing and interpreting nonsense2) The interpretation of regression weights in a main effects model is different than in a model including interactions• regression weights reflect “main effects” in a maineffects model (without interactions)• regression weights reflect “simple effects” in a modelincluding interactionsy’ = bX + aModels with a single quantitative predictora Æ regression constant• expected value of y if x = 0• height of predictor-criterion regression lineb Æ regression weight• expected direction and extent of change in y for a 1-unit increase in x • slope of line0 10 20 30 40 50 60 Å Yy’ = bX + aa+ b0 10 20 30 40 Å Xa = height of lineb = slope of lineGraphing & InterpretingModels with a single quantitative predictory’ = bXcen+ aModels with a single centered quantitative predictora Æ regression constant• expected value of y when x = 0 (x = mean )• height of predictor-criterion regression lineb Æ regression weight• expected direction and extent of change in y for a 1-unit increase in x • slope of predictor-criterion regression lineXcen= X – XmeanX will be Xcenin all of the following modelsy’ = bXcen+ aModels with a single centered quantitative predictorXcen= X – XmeanSo, how do we plot this formula?Simple Æ pick 2 values of xcen, substitute them into the formula to get y’ values and plot the line defined by those two x-y pointsWhat x values?• doesn’t matter -- so keep it simple…• 0 & 1, 0 & 10, 0 & 100, depending on the x-scale• +/- 2 std isn’t as simple, but tells you what x & y ranges are needed on the plotCouple of things to remember…• “0” is the center of the x-axis – X has been centered !!!• the x-axis should extend about +/- 2 Std (include 96% of pop)y’ = 1.5X + 30Models with a single centered quantitative predictor0 10 20 30 40 50 60 70-20 -10 0 10 20 Å XcenFor xcen= 0 Æ 1.6*0 + 30 y’ = 30For xcen= 10 Æ 1.6*10 + 30 y’ = 46If X has mean = 42 & std = 7.5For +2stdÆ 1.6*15 + 30 y’ = 54For -2std Æ 1.6*-15 + 30 y’ = 6Any set of x values will lead to the same plotted line!20 30 40 50 60 Å XWe can even substitute the original x scale back into the graph !!0 10 20 30 40 50 60a (x = mean)+b-20 -10 0 10 20 Å Xcena = ht of lineb= slpof lineGraphing & Interpreting Models with a single centered quantitative predictorXcen= X – Xmeany’ = bXcen+ a0 10 20 30 40 50 60b= 0-20 -10 0 10 20 Å Xcena = ht of lineb= slpof linea (x = mean)Graphing & Interpreting Models with a single centered quantitative predictorXcen= X – Xmeany’ = bXcen+ a-20 -10 0 10 20 Å Xcen0 10 20 30 40 50 60-ba = ht of lineb= slpof linea (x = mean)Graphing & Interpreting Models with a single centered quantitative predictorXcen= X – Xmeany’ = -bXcen+ ay’ = bX + aModels with a single binary predictor coded 1-2a Æ regression constant• expected value of y if x = 0 (no one has this value – all coded 1 or 2)• height of lineb Æ regression weight• expected direction and extent of change in y for a 1-unit increase in x • direction and extent of y mean difference between groups coded 1 & 2• slope of line0 10 20 30 40 50 60y’ = bZ + aControl Tx1 Cx = 1 Tx = 2Z = Tx1 vs. CxPlotting & Interpretingmodels with a single binary predictor coded 1-2+b0 10 20 30 40 50 60y’ = bZ + aControl Tx1 Cx = 1 Tx = 2Z = Tx1 vs. CxPlotting & Interpretingmodels with a single binary predictor coded 1-2b = 00 10 20 30 40 50 60y’ = -bZ + aControl Tx1 Cx = 1 Tx = 2Z = Tx1 vs. CxPlotting & Interpretingmodels with a single binary predictor coded 1-2-by’ = bZ + aa Æ regression constant• expected value of y if z = 0 (the control group)• mean of the control group• height of control groupb Æ regression weight• expected direction and extent of change in y for a 1-unit increase in x • direction and extent of y mean difference between groups coded 0 & 1• group height differenceModels with a single dummy coded binary predictorThe way we’re going to graph this model looks